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<h1>Minimize thermal noise power of an array with arbitrary 2-D geometry</h1>
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<span class="comment">% "Convex optimization examples" lecture notes (EE364) by S. Boyd</span>
<span class="comment">% "Antenna array pattern synthesis via convex optimization"</span>
<span class="comment">% by H. Lebret and S. Boyd</span>
<span class="comment">% (figures are generated)</span>
<span class="comment">%</span>
<span class="comment">% Designs an antenna array such that:</span>
<span class="comment">% - it has unit a sensitivity at some target direction</span>
<span class="comment">% - obeys constraint for minimum sidelobe level outside the beamwidth</span>
<span class="comment">% - minimizes thermal noise power in y (sigma*||w||_2^2)</span>
<span class="comment">%</span>
<span class="comment">% This is a convex problem described as:</span>
<span class="comment">%</span>
<span class="comment">%   minimize   norm(w)</span>
<span class="comment">%       s.t.   y(theta_tar) = 1</span>
<span class="comment">%              |y(theta)| &lt;= min_sidelobe   for theta outside the beam</span>
<span class="comment">%</span>
<span class="comment">% where y is the antenna array gain pattern (complex function) and</span>
<span class="comment">% variables are w (antenna array weights or shading coefficients).</span>
<span class="comment">% Gain pattern is a linear function of w: y(theta) = w'*a(theta)</span>
<span class="comment">% for some a(theta) describing antenna array configuration and specs.</span>
<span class="comment">%</span>
<span class="comment">% Written for CVX by Almir Mutapcic 02/02/06</span>

<span class="comment">% select array geometry</span>
ARRAY_GEOMETRY = <span class="string">'2D_RANDOM'</span>;
<span class="comment">% ARRAY_GEOMETRY = '1D_UNIFORM_LINE';</span>
<span class="comment">% ARRAY_GEOMETRY = '2D_UNIFORM_LATTICE';</span>

<span class="comment">%********************************************************************</span>
<span class="comment">% problem specs</span>
<span class="comment">%********************************************************************</span>
lambda = 1;           <span class="comment">% wavelength</span>
theta_tar = 60;       <span class="comment">% target direction</span>
half_beamwidth = 10;  <span class="comment">% half beamwidth around the target direction</span>
min_sidelobe = -20;   <span class="comment">% maximum sidelobe level in dB</span>

<span class="comment">%********************************************************************</span>
<span class="comment">% random array of n antenna elements</span>
<span class="comment">%********************************************************************</span>
<span class="keyword">if</span> strcmp( ARRAY_GEOMETRY, <span class="string">'2D_RANDOM'</span> )
  <span class="comment">% set random seed to repeat experiments</span>
  rand(<span class="string">'state'</span>,0);

  <span class="comment">% (uniformly distributed on [0,L]-by-[0,L] square)</span>
  n = 36;
  L = 5;
  loc = L*rand(n,2);

<span class="comment">%********************************************************************</span>
<span class="comment">% uniform 1D array with n elements with inter-element spacing d</span>
<span class="comment">%********************************************************************</span>
<span class="keyword">elseif</span> strcmp( ARRAY_GEOMETRY, <span class="string">'1D_UNIFORM_LINE'</span> )
  <span class="comment">% (unifrom array on a line)</span>
  n = 30;
  d = 0.45*lambda;
  loc = [d*[0:n-1]' zeros(n,1)];

<span class="comment">%********************************************************************</span>
<span class="comment">% uniform 2D array with m-by-m element with d spacing</span>
<span class="comment">%********************************************************************</span>
<span class="keyword">elseif</span> strcmp( ARRAY_GEOMETRY, <span class="string">'2D_UNIFORM_LATTICE'</span> )
  m = 6; n = m^2;
  d = 0.45*lambda;

  loc = zeros(n,2);
  <span class="keyword">for</span> x = 0:m-1
    <span class="keyword">for</span> y = 0:m-1
      loc(m*y+x+1,:) = [x y];
    <span class="keyword">end</span>
  <span class="keyword">end</span>
  loc = loc*d;

<span class="keyword">else</span>
  error(<span class="string">'Undefined array geometry'</span>)
<span class="keyword">end</span>

<span class="comment">%********************************************************************</span>
<span class="comment">% construct optimization data</span>
<span class="comment">%********************************************************************</span>
<span class="comment">% build matrix A that relates w and y(theta), ie, y = A*w</span>
theta = [1:360]';
A = kron(cos(pi*theta/180), loc(:,1)') + kron(sin(pi*theta/180), loc(:,2)');
A = exp(2*pi*i/lambda*A);

<span class="comment">% target constraint matrix</span>
[diff_closest, ind_closest] = min( abs(theta - theta_tar) );
Atar = A(ind_closest,:);

<span class="comment">% stopband constraint matrix</span>
ind = find(theta &lt;= (theta_tar-half_beamwidth) | <span class="keyword">...</span>
           theta &gt;= (theta_tar+half_beamwidth) );
As = A(ind,:);

<span class="comment">%********************************************************************</span>
<span class="comment">% optimization problem</span>
<span class="comment">%********************************************************************</span>
cvx_begin
  variable <span class="string">w(n)</span> <span class="string">complex</span>
  minimize( norm( w ) )
  subject <span class="string">to</span>
    Atar*w == 1;
    abs(As*w) &lt;= 10^(min_sidelobe/20);
cvx_end

<span class="comment">% check if problem was successfully solved</span>
disp([<span class="string">'Problem is '</span> cvx_status])
<span class="keyword">if</span> ~strfind(cvx_status,<span class="string">'Solved'</span>)
  <span class="keyword">return</span>
<span class="keyword">end</span>

fprintf(1,<span class="string">'The minimum norm of w is %3.2f.\n\n'</span>,norm(w));

<span class="comment">%********************************************************************</span>
<span class="comment">% plots</span>
<span class="comment">%********************************************************************</span>
figure(1), clf
plot(loc(:,1),loc(:,2),<span class="string">'o'</span>)
title(<span class="string">'Antenna locations'</span>)

<span class="comment">% plot array pattern</span>
y = A*w;

figure(2), clf
ymin = -30; ymax = 0;
plot([1:360], 20*log10(abs(y)), <span class="keyword">...</span>
     [theta_tar theta_tar],[ymin ymax],<span class="string">'r--'</span>,<span class="keyword">...</span>
     [theta_tar+half_beamwidth theta_tar+half_beamwidth],[ymin ymax],<span class="string">'g--'</span>,<span class="keyword">...</span>
     [theta_tar-half_beamwidth theta_tar-half_beamwidth],[ymin ymax],<span class="string">'g--'</span>,<span class="keyword">...</span>
     [0 theta_tar-half_beamwidth],[min_sidelobe min_sidelobe],<span class="string">'r--'</span>,<span class="keyword">...</span>
     [theta_tar+half_beamwidth 360],[min_sidelobe min_sidelobe],<span class="string">'r--'</span>);
xlabel(<span class="string">'look angle'</span>), ylabel(<span class="string">'mag y(theta) in dB'</span>);
axis([0 360 ymin ymax]);

<span class="comment">% polar plot</span>
figure(3), clf
zerodB = 50;
dBY = 20*log10(abs(y)) + zerodB;
plot(dBY.*cos(pi*theta/180), dBY.*sin(pi*theta/180), <span class="string">'-'</span>);
axis([-zerodB zerodB -zerodB zerodB]), axis(<span class="string">'off'</span>), axis(<span class="string">'square'</span>)
hold <span class="string">on</span>
plot(zerodB*cos(pi*theta/180),zerodB*sin(pi*theta/180),<span class="string">'k:'</span>) <span class="comment">% 0 dB</span>
plot( (min_sidelobe + zerodB)*cos(pi*theta/180), <span class="keyword">...</span>
      (min_sidelobe + zerodB)*sin(pi*theta/180),<span class="string">'k:'</span>)  <span class="comment">% min level</span>
text(-zerodB,0,<span class="string">'0 dB'</span>)
text(-(min_sidelobe + zerodB),0,sprintf(<span class="string">'%0.1f dB'</span>,min_sidelobe));
theta_1 = theta_tar+half_beamwidth;
theta_2 = theta_tar-half_beamwidth;
plot([0 55*cos(theta_tar*pi/180)], [0 55*sin(theta_tar*pi/180)], <span class="string">'k:'</span>)
plot([0 55*cos(theta_1*pi/180)], [0 55*sin(theta_1*pi/180)], <span class="string">'k:'</span>)
plot([0 55*cos(theta_2*pi/180)], [0 55*sin(theta_2*pi/180)], <span class="string">'k:'</span>)
hold <span class="string">off</span>
</pre>
<a id="output"></a>
<pre class="codeoutput">
 
Calling Mosek 9.1.9: 1439 variables, 414 equality constraints
   For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 414             
  Cones                  : 342             
  Scalar variables       : 1439            
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 414             
  Cones                  : 342             
  Scalar variables       : 1439            
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 72
Optimizer  - Cones                  : 343
Optimizer  - Scalar variables       : 1099              conic                  : 1099            
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 2628              after factor           : 2628            
Factor     - dense dim.             : 0                 flops                  : 5.53e+06        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   0.0e+00  1.0e+00  3.6e+01  0.00e+00   3.410000000e+01   -1.000000000e+00  1.0e+00  0.01  
1   5.3e-14  9.4e-01  3.7e+01  1.11e+01   1.440560643e+01   -1.338598654e+00  9.4e-01  0.02  
2   1.6e-13  3.5e-01  5.1e+00  4.61e+00   1.884111089e+00   -4.827843861e-01  3.5e-01  0.02  
3   2.3e-13  8.4e-02  4.9e-01  1.61e+00   6.065250993e-03   -4.847201593e-01  8.4e-02  0.03  
4   4.2e-13  6.3e-02  3.2e-01  9.45e-01   -1.485235974e-01  -5.299185532e-01  6.3e-02  0.03  
5   8.8e-13  2.4e-02  6.8e-02  9.82e-01   -4.472091476e-01  -5.976390060e-01  2.4e-02  0.03  
6   1.7e-12  2.0e-03  6.5e-04  1.01e+00   -6.312972273e-01  -6.443621841e-01  2.0e-03  0.03  
7   3.6e-11  1.0e-03  2.3e-04  1.00e+00   -6.412479614e-01  -6.479375854e-01  1.0e-03  0.04  
8   2.9e-11  1.7e-04  1.4e-05  9.96e-01   -6.498607255e-01  -6.509727401e-01  1.7e-04  0.04  
9   5.6e-11  9.2e-06  1.8e-07  1.00e+00   -6.515133315e-01  -6.515725485e-01  9.2e-06  0.04  
10  1.2e-09  1.5e-06  1.1e-08  1.00e+00   -6.515926398e-01  -6.516020441e-01  1.5e-06  0.04  
11  1.1e-09  3.0e-08  3.3e-11  1.00e+00   -6.516072068e-01  -6.516073994e-01  3.0e-08  0.05  
12  1.7e-09  3.9e-09  1.6e-12  1.00e+00   -6.516074744e-01  -6.516074993e-01  3.9e-09  0.05  
13  2.0e-09  1.3e-09  4.1e-17  1.00e+00   -6.516075141e-01  -6.516075141e-01  6.0e-12  0.05  
Optimizer terminated. Time: 0.06    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -6.5160751406e-01   nrm: 3e+00    Viol.  con: 4e-10    var: 0e+00    cones: 0e+00  
  Dual.    obj: -6.5160751430e-01   nrm: 7e-01    Viol.  con: 0e+00    var: 8e-11    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.06    
    Interior-point          - iterations : 13        time: 0.05    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.651608
 
Problem is Solved
The minimum norm of w is 0.65.

</pre>
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<img src="ant_array_min_therm_noise__01.png" alt=""> <img src="ant_array_min_therm_noise__02.png" alt=""> <img src="ant_array_min_therm_noise__03.png" alt=""> 
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